Properties of the Determinant I

Let us call the first matrix $A$ and the second $B$ .

We see we can get $B$ from $A$ by applying the following operations:

1. $R_1 \rightarrow 2R_1$
2. $R_3 \rightarrow 2R_3$
3. $C_1 \rightarrow 2C_1$
4. $C_3 \rightarrow 2C_3$

Where $R_n \rightarrow kR_n$ multiplies every element in row $n$ by $k$ , and Where $C_n \rightarrow kC_n$ multiplies every element in column $n$ by $k$ . These operations can be represented by the matrices:

$E_1 = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $E_2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}$ .

Where pre-multipying is a row operation and post-multiplying is a column operation.

Then $B = E_1E_2AE_1E_2$ , so $\det(B) = \det(E_1)^2\det(E_2)^2\det(A) = 2^2 \cdot 2^2 \cdot 16 = 256$ .

If we use permutations ("Sarrus' Rule") to find the determinant, we see that every term is multiplied by 16, $16a_1a_5a_9+...-16a_3a_5a_7$ , so that the determinant we seek is $16^2=\boxed{256}$ .